Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3+2x^2+3; x-1

Verified step by step guidance

1. Identify the divisor from the second polynomial, which is \( x - 1 \). The root to test using the factor theorem is \( x = 1 \).
2. Set up synthetic division using the coefficients of the first polynomial \( x^3 + 2x^2 + 0x + 3 \), which are \( 1, 2, 0, 3 \).
3. Perform synthetic division by bringing down the leading coefficient (1) and multiply it by the root (1), then add to the next coefficient.
4. Continue the synthetic division process: multiply the result by the root and add to the next coefficient, repeating until all coefficients are processed.
5. Check the remainder: if the remainder is 0, then \( x - 1 \) is a factor of the polynomial. If not, it is not a factor.

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This means that if you substitute c into the polynomial and the result is zero, then (x - c) is a factor of the polynomial. This theorem is essential for determining factors of polynomials and is often used in conjunction with synthetic division.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of arithmetic operations to find the quotient and remainder. This technique is faster and more efficient than traditional long division, especially for polynomials of higher degrees.

Polynomials

A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers and coefficients. The general form of a polynomial in one variable x is a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n are constants. Understanding the structure and properties of polynomials is crucial for applying the Factor Theorem and synthetic division effectively.